scholarly journals Survey on the geometric Bogomolov conjecture

2017 ◽  
pp. 137-193
Author(s):  
Kazuhiko Yamaki
Keyword(s):  
Bogomolov Conjecture

scholarly journals The geometric Bogomolov conjecture

2020 ◽  
Author(s):  
Serge Cantat ◽  
Ziyang Gao ◽  
Philipp Habegger ◽  
Junyi Xie
Keyword(s):  
Bogomolov Conjecture

scholarly journals Effective results for points on certain subvarieties of tori

2009 ◽  
Vol 147 (1) ◽  
pp. 69-94 ◽  
Author(s):  
ATTILA BÉRCZES ◽  
KÁLMÁN GYŐRY ◽  
JAN-HENDRIK EVERTSE ◽  
CORENTIN PONTREAU
Keyword(s):  
Number Field ◽  
Finite Rank ◽  
Upper Bounds ◽  
Accurate Description ◽  
Precise Form ◽  
Small Height ◽  
Bogomolov Conjecture ◽  
Special Classes

AbstractThe combined conjecture of Lang-Bogomolov for tori gives an accurate description of the set of those pointsxof a given subvarietyof$\mathbb{G}_{\bf m}^N(\oQ )=(\oQ^*)^N$, that with respect to the height are “very close” to a given subgroup Γ of finite rank of$\mathbb{G}_{\bf m}^N(\oQ)$. Thanks to work of Laurent, Poonen and Bogomolov, this conjecture has been proved in a more precise form.In this paper we prove, for certain special classes of varieties, effective versions of the Lang-Bogomolov conjecture, giving explicit upper bounds for the heights and degrees of the pointsxunder consideration. The main feature of our results is that the points we consider do not have to lie in a prescribed number field. Our main tools are Baker-type logarithmic forms estimates and Bogomolov-type estimates for the number of points on the varietywith very small height.

scholarly journals A consequence of the relative Bogomolov conjecture

2021 ◽  
Author(s):  
Vesselin Dimitrov ◽  
Ziyang Gao ◽  
Philipp Habegger
Keyword(s):  
Bogomolov Conjecture

scholarly journals Generic rank of Betti map and unlikely intersections

2020 ◽  
Vol 156 (12) ◽  
pp. 2469-2509
Author(s):  
Ziyang Gao
Keyword(s):  
Shimura Varieties ◽  
Connected Subset ◽  
Geometric Criterion ◽  
The Real ◽  
Formula One ◽  
Abelian Scheme ◽  
Generic Rank ◽  
Irreducible Variety ◽  
Simply Connected ◽  
Bogomolov Conjecture

Let $\mathcal {A} \rightarrow S$ be an abelian scheme over an irreducible variety over $\mathbb {C}$ of relative dimension $g$. For any simply-connected subset $\Delta$ of $S^{\mathrm {an}}$ one can define the Betti map from $\mathcal {A}_{\Delta }$ to $\mathbb {T}^{2g}$, the real torus of dimension $2g$, by identifying each closed fiber of $\mathcal {A}_{\Delta } \rightarrow \Delta$ with $\mathbb {T}^{2g}$ via the Betti homology. Computing the generic rank of the Betti map restricted to a subvariety $X$ of $\mathcal {A}$ is useful to study Diophantine problems, e.g. proving the geometric Bogomolov conjecture over char $0$ and studying the relative Manin–Mumford conjecture. In this paper we give a geometric criterion to detect this rank. As an application we show that it is maximal after taking a large fibered power (if $X$ satisfies some conditions); it is an important step to prove the bound for the number of rational points on curves (Dimitrov et al., Uniformity in Mordell–Lang for Curves, Preprint (2020), arXiv:2001.10276). Another application is to answer a question of André, Corvaja and Zannier and improve a result of Voisin. We also systematically study its link with the relative Manin–Mumford conjecture, reducing the latter to a simpler conjecture. Our tools are functional transcendence and unlikely intersections for mixed Shimura varieties.

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